Filter banks for enhancing signals using oversampled subband transforms

ABSTRACT

For subband decomposition of a d-dimensional input signal (S) into a number K of subband components (F 1 -F 4 ), a filter bank has a filtering module ( 801 ) transforming the input signal (S) into 2 d  components including a low-frequency component (L) and 2 d −1 higher-frequency components (F 1 ), The 2 d −1 higher-frequency components are oversampled, typically by a factor 2, compared to the low-frequency component. The low-frequency component can be further decomposed by means of another filtering module having a similar structure, and the process can be iterated over any number of scales. The reconstruction filter bank has a symmetric structure, with filtering modules adapted to the oversampling of the higher-frequency components. Such filter banks are well suited to various enhancement processing applied to the subband components such as thresholding, reduction of compression distortion, reduction of measurement noise, sharpness enhancement.

BACKGROUND OF THE INVENTION

The present invention relates to signal processing technology and moreparticularly to techniques for enhancing signals, such as by removingnoise, artifacts or blur, enhancing sharpness, etc.

It is generally applicable to the enhancement of d-dimensional signals,where d is some positive integer (d≧1). Audio signals are examples ofone-dimensional signals. Images are examples of two-dimensional signals.Three-dimensional signals may correspond to, e.g., video sequences orthree-dimensional blocks of data such as seismic data or medical imagingdata. Enhancement is distinguished from compression which eithermaintains or degrades the signal in order to construct a compact binarycode representing it.

Signal enhancement or restoration is a process that improves an inputdigital signal by removing noise components or by suppressing existingdistortions introduced by some prior transformation or degradationprocess such as blurring or signal compression process. Sharpening thesignal by removing blur is a form of signal restoration as well asremoval of compression artifacts or any additive noise.

Many efficient signal enhancement methods are implemented by means offilter banks that transform the signal into a set of subband signals.Wavelet and wavepacket transforms are examples of such subbandtransformations. Typically, the transformed coefficients are thenprocessed with simple non-linear amplification or attenuation operatorssuch as soft or hard thresholding operators or block thresholdingoperators, as described in D. Donoho and I. Johnstone “Ideal spatialadaptation via wavelet shrinkage”, Biometrika, vol. 81, pp. 425-455,December 1994. An inverse subband transform is then used reconstruct anenhanced signal from the processed subband coefficients.

The filter banks used for subband decomposition implement orthogonal orbiorthogonal subband transforms with critically-sampled filter banks, asdescribed in M. Vetterli and C. Herley, “Wavelets and filter banks,theory and design”, IEEE Transactions on Signal Processing, vol. 40, no.9, pp. 2207-2232, September 1992. The inverse subband transform isperformed by means of perfect reconstruction filters. For a signal ofsize N, the total number of subband coefficients is also equal to N. Thememory size and the number of operations required by critically-sampledfilter bank transforms is proportional to N. Orthogonal or biorthogonalwavelet transforms are instances of such transforms. These transformsare computationally very efficient but the subsampling incorporated inthe filter bank introduces grid artefacts on the reconstruction. This isparticularly visible with a Haar wavelet transform where thereconstructed image has block artifacts (see FIG. 14( c)).

Translation-invariant subband transforms have been introduced to avoidsuch grid artifacts. A translation-invariant subband transform isimplemented by means of a filter bank using an “a trou algorithm”without any subsampling, with zeros incorporated between filtercoefficients, as described in M. J. Shensa “The discrete wavelettransform: wedding the à trous and Mallat algorithms”, IEEE Transactionson Signal Processing, vol. 40, no. 10, pp. 2464-2482, October 1992.Translation-invariant subband transforms remove the grid artifacts andgenerally improve the peak signal-to-noise ratio (PSNR) of enhancementsystems compared to equivalent critically-sampled subband transforms.However, they require much larger memory size and computationalcomplexity.

Compared to a critically subsampled filter bank, a translation invariantfilter bank increases the memory size and the number of operations by afactor that is approximately equal to the number of frequency subbands.For a wavelet transform computed over J scales, this factor is J+1 forone-dimensional signals, 3J+1 for two-dimensional signals and 7J+1 forthree-dimensional signals. The number of scales J is larger than 3 inmany applications. For other wavelet packet subband transforms, thesefactors are often larger than for a wavelet transform.

There is a need for subband transform schemes that attenuate gridartefacts with a smaller computational and memory cost thantranslation-invariant subband transforms. This is particularly importantfor large size signals such as images and videos, for real-timeprocessing applications.

SUMMARY OF THE INVENTION

Filter banks for subband decomposition and reconstruction of ad-dimensional signals are proposed (d being an integer at least equal to1), as well as a signal enhancement system making use of such filterbanks.

On the input side, the filter bank decomposes an input signal into anumber K of subband components for processing. It comprises a filteringmodule for transforming the input signal into 2^(d) components includinga low-frequency component and 2^(d)−1 higher-frequency components. These2^(d)−1 higher-frequency components are oversampled compared to thelow-frequency component.

In an embodiment, the low-frequency component is downsampled by a factor2^(d) compared to the input signal. The 2^(d)−1 higher-frequencycomponents may then include; a highest-frequency component having asmany samples as the input signal; and, if d>1, further componentsoversampled by respective factors 2^(i) compared to the low-frequencycomponent, each i being an integer greater than 0 and smaller than d.

The subband decomposition can be performed over multiple scales. Thefilter bank then comprises filtering modules organized in a tree ofdepth J, J being the number of scales of the subband decomposition. Eachof the filtering modules can be arranged to transform a respective inputsignal into 2^(d) respective components including a low-frequencycomponent and 2^(d)−1 higher-frequency components oversampled comparedto the respective low-frequency component. The input signal of thefilter bank is then the respective input signal of a filtering module atthe root of the tree.

In an embodiment using a wavelet type of transform, the total number offiltering modules in the tree will typically be J, with the J modulesarranged in cascade, the low-frequency component from the j-th filteringmodule being the input signal to the (j+1)-th filtering module, for j=1,. . . , J−1. In such a J-scale embodiment, the K subband components mayinclude the low-frequency component from the J-th filtering module andthe 2^(d)−1 higher-frequency components from each one of the J cascadedfiltering modules. The low-frequency component from the J-th filteringmodule is typically downsampled by a factor 2^(d.J) compared to theinput signal, and the K subband components include:

-   -   the low-frequency component from the J-th filtering module;    -   a highest-frequency component oversampled by a factor 2^(d)        compared to said low-frequency component; and    -   if at least one of d and J is greater than 1, further components        oversampled by respective factors 2^(i) compared to said        low-frequency component, each i being an integer greater than 0        and smaller than d.J.

If a wavelet packet type of transform is used, there will generally bemore than one filtering module per level in the tree of thedecomposition filter bank.

In an embodiment, each filtering module is made of 2^(d)−1 filteringunits arranged in a binary tree having d levels. For 1≦i≦d, the i-thlevel in the tree has 2^(i−1) filtering units each receiving arespective input signal and producing a respective output low-frequencysignal and a respective output high-frequency signal oversampledcompared to said respective output low-frequency signal. The inputsignal of the filtering module is the input signal of the filtering unitof the first level in the tree. For any i>1, the output low-frequencyand high-frequency signals from the 2^(i−2) filtering units of the(i−1)-th level in the tree are the respective input signals of the2^(i−1) filtering units of the i-th level in the tree. The 2^(d)respective components from the filtering module are the outputlow-frequency and high-frequency signals from the 2^(d−1) filteringunits of the d-th level in the tree.

On the output side, the filter bank reconstructs an output signal from Kprocessed subband components. It comprises a filtering module forgenerating the output signal from 2^(d) components obtained from the Ksubband components, including a low-frequency component and 2^(d)−1higher-frequency components. These 2^(d)−1 higher-frequency componentsare oversampled compared to said low-frequency component.

In an embodiment of the reconstruction filter bank, the output signal isoversampled by a factor 2^(d) compared to the low-frequency component,and the 2^(d)−1 higher-frequency components include a highest-frequencycomponent having as many samples as the output signal and, if d>1,further components oversampled by respective factors 2^(i) compared tothe low-frequency component, each i being an integer greater than 0 andsmaller than d.

When the subband components supplied to the reconstruction filter bankresult from a wavelet decomposition over multiple scales (J), the filterbank may include J cascaded filtering modules. Each of the filteringmodules is arranged to transform 2^(d) respective input components,including a low-frequency component and 2^(d)−1 higher-frequencycomponents oversampled compared to the respective low-frequencycomponent, into a respective output signal. For j=1, . . . , J−1, theoutput signal from the j-th filtering module is then the respectivelow-frequency component supplied to the (j+1)-th filtering module, whilethe output signal from the J-th filtering module is the output signal ofthe filter bank. In such a J-scale embodiment, the 2^(d)−1higher-frequency components supplied to each of the J cascaded filteringmodules and the low-frequency component supplied to the first filteringmodule can be components from the input K subband components. The outputsignal is typically oversampled by a factor 2^(d.J) compared to thelow-frequency component supplied to the first filtering module, and theK subband components include:

-   -   the low-frequency component supplied to the first filtering        module;    -   a highest-frequency component oversampled by a factor 2^(d)        compared to the low-frequency component supplied to the first        filtering module; and    -   if at least one of d and J is greater than 1, further components        oversampled by respective factors 2^(i) compared to the        low-frequency component supplied to the first filtering module,        each i being an integer greater than 0 and smaller than d.J.

For j=1, . . . , J−1, the output signal from the j-th filtering moduleis then the respective low-frequency component supplied to the (j+1)-thfiltering module, while the output signal from the J-th filtering moduleis the output signal of the filter bank.

When the subband components supplied to the reconstruction filter bankresult from a wavelet packet decomposition over multiple scales (J), thefilter bank may include filtering modules arranged in a tree of depth J.Each of the filtering modules is arranged to transform 2^(d) respectiveinput components, including a low-frequency component and 2^(d)−1higher-frequency components oversampled compared to the respectivelow-frequency component, into a respective output signal.

In an embodiment of the reconstruction filter bank, each filteringmodule is made of 2^(d)−1 filtering units arranged in a binary treehaving d levels. For 1≦i≦d, the i-th level in the tree has 2^(d−i)filtering units each receiving a respective input low-frequency signaland a respective input high-frequency signal oversampled compared to therespective input low-frequency signal and producing a respective outputsignal. The 2^(d) respective input components supplied to the filteringmodule are distributed as respective input low-frequency andhigh-frequency signals to the 2^(d−1) filtering units of the first levelin the tree. For any i>1, the respective input low-frequency andhigh-frequency signals of the 2^(d−i) filtering units of the i-th levelin the tree are the respective output signals from the 2^(d−i+1)filtering units of the (i−1)-th level in the tree. The output signal ofthe filtering module is the output signal of the filtering unit of thed-th level in the tree.

One or more of the filtering units can be structured with:

-   -   a first branch for filtering an upsampled version of the        respective input low-frequency signal of the filtering unit to        form a first component signal;    -   a second branch for filtering a first modified version of the        respective input high-frequency signal of the filtering unit, in        which the odd samples are replaced by zeroes, to form a second        component signal;    -   a first adder to form a first partially reconstructed signal as        a sum of the first and second component signals;    -   a third branch for filtering the partially reconstructed signal        to form a third component signal;    -   a fourth branch for filtering a second modified version of the        respective input high-frequency signal of the filtering unit, in        which the even samples are replaced by zeroes, to form a fourth        component signal;    -   a second adder to form a second partially reconstructed signal        as a sum of the third and fourth component signals; and    -   a combiner to produce the respective output signal of the        filtering unit as a combination of the first and second        partially reconstructed signals.

The K processed subband components being obtained from a subbanddecomposition involving a low-pass filter h₁ and a high-pass filter g₁,the filtering performed in the first and second branches is preferablybased on respective filters h₂ and g₂ such that the filter pairs {h₁,g₁} and {h₂, g₂} verify a perfect reconstruction property. The filteringperformed in the third branch is based on a combination of filters h₁and h₂, while the filtering performed in the fourth branch is also basedon filter g₂.

Alternatively, one or more of the filtering units can be structuredwith:

-   -   a first filter for generating a first even sub-component from        the respective input low-frequency signal of the filtering unit;    -   a second filter for generating a first odd sub-component from        the respective input low-frequency signal of the filtering unit;    -   a third filter for generating a second even sub-component from a        first downsampled version of the respective input high-frequency        signal of the filtering unit;    -   a fourth filter for generating a second odd sub-component from        said first downsampled version of the input high-frequency        signal;    -   a fifth filter for generating a third even sub-component from a        second downsampled version of the respective input        high-frequency signal of the filtering unit;    -   a sixth filter for generating a third odd sub-component from        said second downsampled version of the input high-frequency        signal;    -   a combiner to produce the respective output signal of the        filtering unit having even components respectively obtained as a        sum of the first, second and third even sub-components and as a        sum of the first, second and third odd sub-components.

The K processed subband components being obtained from a subbanddecomposition involving a high-pass filter g₁ and a low-pass filter h₁implemented in a polyphase filtering arrangement, the first, second,third, fourth, fifth and sixth filters are defined from the low-pass andhigh-pass filters h₁, g₁ and from respective inverse filters h₂ and g₂such that the filter pairs {h₁, g₁} and {h₂, g₂} verify a perfectreconstruction property.

The above-disclosed filter banks for subband decomposition andreconstruction have been studied and it was found that they caneliminate or at least reduce substantially grid or block artifactsintroduced by known critically-sampled filter banks. This is achieved atthe cost of an increase of complexity in terms of computation and memoryrequirements. Yet, the increase of complexity is much smaller than it isfor known alternatives including translation-invariant filter banks.

For a wavelet transform over J scales of a one-dimensional signal, atwice oversampled filter bank increases by about a factor 2 the memoryand number of operations relatively to a critically-sampled filter bank,whereas this factor is J+1 for a translation-invariant transform. Forimages (2D signals), the memory and number of operations increase by afactor about 3 for a twice oversampled filter bank as opposed to 3J+1for a translation-invariant filter bank. For a video (3D signal), thisincreasing factor is approximately 4 for a twice oversampled filter bankas opposed to 7J+1 for a translation invariant filter bank. For typicalvalues of J≧3, twice oversampled filter banks thus reduce the memory andcomputations by an important factor compared to translation-invariantfilter banks, for signal enhancement systems or in other applications ofsubband decomposition/reconstruction.

A signal enhancement system according to the invention comprises:

-   -   a first filter bank as disclosed above for subband decomposition        of an input signal into a number K of subband components;    -   a subband enhancement module for processing the K subband        components from the first filter bank and forming K enhanced        subband component; and    -   a second filter bank as disclosed above for reconstruction of an        output signal from the K enhanced subband components.

The subband enhancement module may be arranged to perform a processingselected from such processing as thresholding, reduction of compressiondistortion, reduction of measurement noise, sharpness enhancement.

The signal enhancement system is implemented using filter bankdecomposition with an oversampling and an inverse filter bankreconstruction. The oversampling is typically by a factor two. The twiceoversampled filter bank removes nearly all grid artifacts produced by acritically-sampled filter bank, with a significantly a lower memory andcomputational cost than a translation invariant filter bank.

The system can be implemented by means of either hardware of software.In the hardware case, the important reduction of the memory sizerequirement is an important factor for cost reduction.

BRIEF DESCRIPTION OF THE DRAWINGS

The foregoing and other objects of this invention, the various featuresthereof, as well as the invention itself, may be more fully understoodfrom the following description, when read together with the accompanyingdrawings in which:

FIG. 1 is a block diagram of signal enhancement system;

FIGS. 2 and 3 are block diagrams of a conventional critically-sampledfiltering unit for subband decomposition of a one-dimensional signal andof a corresponding inverse filtering unit for signal reconstruction;

FIGS. 4 and 5 are block diagrams of exemplary configurations of a twiceoversampled subband filtering unit for decomposition of aone-dimensional signal and of a corresponding inverse filtering unit forsignal reconstruction;

FIGS. 6 and 7 are block diagrams of polyphase configurations for a twiceoversampled subband filtering unit for decomposition of aone-dimensional signal and for a corresponding inverse filtering unitfor signal reconstruction;

FIGS. 8 and 9 are block diagrams of exemplary filter banks formultiscale subband decomposition and reconstruction of one-dimensionalsignals;

FIGS. 10 and 11 are block diagrams of exemplary filter banks formultiscale subband decomposition and reconstruction of a two-dimensionalimage;

FIGS. 12 and 13 are block diagrams of exemplary filter banks formultiscale subband decomposition and reconstruction of athree-dimensional video;

FIG. 14 shows (a) an example of original image, (b) the same imagecontaminated by Gaussian additive white noise, (c) an image obtained bythresholding orthogonal Haar coefficients on 4 scales and (d) an imageobtained by thresholding twice oversampled Haar coefficients.

DESCRIPTION OF EMBODIMENTS

The formalism of the mathematical expressions in the following is wellknown to those skilled in the art. We write f*h[n]=Σ_(m)f[n−m].h[m] theone dimensional convolution, also called filtering, of a signal f[n]with a filter h[n]. The z-transform of f is

${\hat{f}(z)} = {\sum\limits_{n = {- \infty}}^{+ \infty}\; {{f\lbrack n\rbrack} \cdot {z^{- n}.}}}$

For a signal or filter f[n], a polyphase filtering separates the evencomponent corresponding to even samples f_(e)[n] and the odd componentcorresponding to odd samples f_(o)[n], which are defined by {circumflexover (f)}(z)={circumflex over (f)}_(e)(z²)+z⁻¹.{circumflex over(f)}_(o)(z²)

FIG. 1 shows a system exemplifying the present invention. It takes ininput a d-dimensional digital signal S. This input signal S is specifiedby its values over a d-dimensional sampling grid. Each sampling point iswritten n=(n₁, . . . , n_(d)), where n₁, n₂, . . . , n_(d) are integers,and the corresponding signal value is denoted S[n]. Audio signals areexamples of 1-dimensional signals, images are examples of 2-dimensionalsignals, and video image sequences are examples of 3-dimensionalsignals. The system of FIG. 1 outputs an enhanced signal S that isdefined on the same sampling grid as the input.

The system of FIG. 1 has a twice oversampled subband filter (TOSF) bank101 receiving the original signal S. The filter bank 101 computes atwice oversampled subband transform and outputs twice oversampledsubband signals (F_(k))_(1≦k≦K) that carry the signal information overdifferent frequency subbands. The number K of frequency subbands dependsupon the type of subband transform.

A subband enhancement module 102 receives the K subband signals(F_(k))_(1≦k≦K) and outputs enhanced subband signals ( F _(k))_(1≦k≦K)using any state of the art enhancement operators. The enhancementcalculation is performed according to a particular application. Noisereduction, reduction of block artifacts produced by compressionalgorithms, sharpness enhancement and suppression of blur are examplesof applications.

In an exemplary embodiment for noise reduction, the subband enhancementcan be implemented by a thresholding operator that sets to zero ordecreases the amplitude of all coefficients below a threshold value thatis proportional to the noise variance. In another exemplary embodiment,coefficients are selected among local maxima of the subband coefficientsand the selected coefficients are set to zero. It yet another exemplaryembodiment, the module 102 implements a block thresholding method thatattenuates the coefficient values depending upon the amplitude ofneighboring subband coefficients. In yet another embodiment, the module102 enhances the high frequencies of the signal by combining athresholding operator and an amplification operator to amplifycoefficients above a threshold.

The system of FIG. 1 further includes a twice oversampled inversesubband filter (TOISF) bank 103 receiving the K enhanced subband signals( F _(k))_(1≦k≦K) and outputting the enhanced signal S by inverting thetwice oversampled subband transform.

A TOSF filter bank is implemented with a cascade of filtering andsubsampling using perfect reconstruction one-dimensional filters of thesame kind as used in a critically-sampled filter bank (see, e.g., M.Vetterli and C. Herley, “Wavelets and filter banks, theory and design”,IEEE Transactions on Signal Processing, vol. 40, no. 9, pp. 2207-2232,September 1992 A two-channel critically-sampled perfect reconstructionfilter bank is known to be defined by two pairs of filters {h₁[n],g₁[n]} and {h₂[n], g₂[n]} whose z-transforms satisfy:

ĥ ₁(−z).ĥ ₂(z)+ĝ ₁(−z).ĝ ₂(z)=0

and

ĥ ₁(z).ĥ ₂(z)+ĝ ₁(z).ĝ ₂(z)=2.

The filters h₁ and h₂ are low-pass filters whereas g₁ and g₂ arehigh-pass filters. Cohen-Daubechies 7/9 and 5/3 biorthogonal perfectreconstruction filters are examples of finite impulse response filters.Conjugate mirror filters are examples of perfect reconstruction filtersfor which h₂[n]=h_(i)[−n] and g₂[n]=g₁[−n]. Daubechies orthogonalfilters are examples of conjugate mirror filters. Haar filters are yetanother example of conjugate mirror filter, in whichĥ₁(z)=(1+z)/√{square root over (2)}, ĝ₁(z)=(1−z)/√{square root over(2)}, ĥ₂(z)=(1+z⁻¹)/√{square root over (2)} and ĝ₂(z)=(1−z⁻¹)√{squareroot over (2)}.

FIG. 2 illustrates an exemplary configuration of a conventionalcritically-sampled subband filtering unit for one-dimensional signals.The input one-dimensional signal A[n] is convolved with theone-dimensional low-pass filter 201 whose impulse response is h₁[n]. Adownsampler 203 receives the low-pass component from filter 201 andoutputs one sample out of two received samples, A*h₁[2n]. Similarly, theinput signal A[n] is convolved with the high-pass filter 202 whoseimpulse response is g₁[n], and a downsampler 204 outputs the evensamples A*g₁[2n]. Each of the resulting low frequency signalL[n]=A*h₁[2n] and high frequency signal H[n]=A*g₁[2n] have twice lesssamples than A[n].

FIG. 3 illustrates an exemplary configuration of a conventionalcritically-sampled subband inverse filtering unit for one-dimensionalsignals. An upsampler 301 inserts zeros in between each sample of aninput low frequency signal L[n], i.e. outputs L ₁[n]= L[n/2] if n iseven and L ₁[n]=0 if n is odd. A filter 303 performs a convolution of L₁[n] with h₂[n] and outputs L ₂[n]= L ₁*h₂[n]. Similarly, an upsampler302 inserts zeros in between each sample of an input high frequencysignal H[n], i.e. outputs H ₁[n]= H[n/2] if n is even and H ₁[n]=0 if nis odd. A filter 304 performs a convolution of H ₁[n] with g₂[n] andoutputs H ₂[n]= H ₁*g₂[n]. An adder 305 produces the output signalĀ[n]of the inverse filtering unit as Ā[n]= L ₂[n]+ H ₂[n].

With perfect reconstruction filters, if the input signals in FIG. 3 areequal to the output signals in FIG. 2, i.e. if L=L and H=H, then theoutput of FIG. 3 is equal to the input of FIG. 2, up to computationalprecision: Ā=A. This is a perfect signal reconstruction property.

Many possibilities are known to those skilled in the art for the choiceof perfect reconstruction filters and efficient algorithms to implementthe convolutions in filters 201, 202, 303 and 304, including liftingschemes, with appropriate boundary treatments of convolutions at thesignal extremities, while retaining the perfect signal reconstructionproperty.

FIG. 4 illustrates an exemplary configuration of a TOSF filtering unit400 usable in the present invention. It includes a low-pass filter 401for convolving the input signal A[n] with the one-dimensional low-passfilter h₁[n], and a downsampler 403 receiving the low-pass componentfrom filter 401 and outputting L[n]=A*h₁[2n]. Filter 401 and downsampler403 can be of the same type as the low-pass filter 201 and downsampler203 described with reference to FIG. 2. The input signal A[n] is alsoconvolved in a high-pass filter 402 having an impulse response g₁[n],which may be identical to the above-described high-pass filter 202.However, no downsampling is applied to the output signal H[n]=A*g₁[n] offilter 402.

The resulting low-pass signal L[n]=A*h₁[2n] has approximately twicefewer samples than A[n], up to one coefficient that depends upon knownborder treatments, whereas the high-pass signal H[n]=A*g₁[n] hasapproximately as many samples as A[n], up to one coefficient that alsodepends upon border treatments.

FIG. 5 illustrates an exemplary configuration of a TOISF filtering unit500 usable in the present invention. Its low frequency input signal L[n]is processed by an upsampler 501 which may be identical to theabove-described upsampler 301. The upsampler 501 outputs L ₁[n] L[n/2]if n is even and L ₁[n]=0 if n is odd. A filter 502, which may beidentical to the above-described filter 301, performs a convolution of L₁[n] with h₂[n] and outputs L ₂[n]= L ₁*h₂[n].

The high frequency input signal H[n] of the TOISF unit 500 is processedby two downsamplers 503, 505. Downsampler 503 outputs the even samplesof H[n], while downsampler 505 outputs the odd samples, the samplessubmitted to downsampler 505 being previously shifted by −1 by the shiftmodule 504. Zeros are inserted by an upsampler 506 between the evensamples output by downsampler 503, while zeros are inserted by anotherupsampler 507 between the odd samples output by downsampler 505. Afilter 508 calculates the convolution between the output of the evenupsampler 506 and g₂[n]. An adder 509 receives the respective outputs offilters 502 and 508 to produce a partially reconstructed signal Ā₁.

A further filter 510 calculates the convolution between the partiallyreconstructed signal Ā₁ and h₁[n]. The output of filter 510 is shiftedby −1 by shifting module 511 and downsampled by a factor 2 by adownsampler 512. Zeroes are then inserted every two samples of theoutput of downsampler 512 by an upsampler 513 whose output is shifted by+1 by a shifting module 514 before being applied to a further filter 515whose impulse response is h₂[n]. The output of the upsampler 507 isshifted by +1 by the shift module 516, and another filter 517 calculatesthe convolution between the output of the shift module 516 and g₂[n]. Afurther adder 518 receives the respective outputs of filters 515 and 517to produce a second partially reconstructed signal Ā₂.

The reconstructed output signal Ā is an average of the two partiallyreconstructed signals Ā₁, Ā₂ weighted by a mixing weight a such that0<a<1:

Ā=a.Ā ₁+(1−a).Ā ₂

This combination is illustrated in FIG. 5 by the multipliers 519-520 andthe adder 521. The mixing weight a should not be equal to 1 but canotherwise be chosen arbitrarily between 0 and 1. In an embodiment, a istaken as equal to ½. This TOISF filtering has a perfect reconstructionproperty, which means that if the input signals L, H in FIG. 5 are equalto the output signals L, H in FIG. 4, ( L=L and H=H), then the outputsignal Ā of FIG. 5 is equal to the input signal A of FIG. 4: Ā=A.

The operations in FIG. 5 can be reconfigured more efficiently by meansof well known filtering techniques. In particular, it is readilyapparent that successive filtering and subbsampling operations can beconcatenated into single steps.

FIGS. 6-7 illustrate alternative embodiments of TOSF and TOISF filteringunits usable in the present invention, based on polyphase filtering.

In the TOSF unit 600 of FIG. 6, the even and odd samples of the inputsignal A are separated by the downsamplers 601, 603, the samplessubmitted to downsampler 603 being previously shifted by −1 by the shiftmodule 602. A first low-pass filter 604 calculates the convolutionbetween the even samples of A, extracted by downsampler 601, and theeven components h_(1e)[n]=h₁[2n] of the low-pass filter h₁. A secondlow-pass filter 605 calculates the convolution between the odd samplesof A, extracted by downsampler 603, and the odd componentsh_(1o)[n]=h₁[2n−1] of the low-pass filter h₁. An adder 606 receives therespective outputs of filters 604 and 605 to produce the low-frequencysignal L. The high frequency signal H is obtained by processing theinput signal A in the high-pass filter 607 having g₁[n] as an impulseresponse, with no downsampling.

The TOISF unit 700 of FIG. 7 includes perfect reconstruction polyphasefilters 704-709 having respective impulse responses x₁[n]-x₆[n] computedfrom the perfect reconstruction filters and the mixing weight a. In apreferred embodiment, these filters are defined by:

{circumflex over (x)} ₁(z)=a.ĥ _(2e)(z)+(1−a).z ⁻¹ .ĥ _(2o)(z).ĥ′_(o)(z)

{circumflex over (x)} ₂(z)=a.ĥ _(2o)(z)+(1−a).ĥ _(2e)(z).ĥ _(2e)(z).ĥ′_(o)(z)

{circumflex over (x)} ₃(z)=a.ĝ _(2e)(z)+(1−a)⁻¹ .ĥ ^(2o)(z).ĝ′ _(o)(z)

{circumflex over (x)} ₄(z)=a.ĝ _(2o)(z)+(1−a).ĥ _(2e)(z).ĝ′ _(o)(z)

{circumflex over (x)} ₅(z)=(1−a).z ⁻¹ .ĝ _(2o)(z)

{circumflex over (x)} ₆(z)=(1−a).ĝ _(2e)(z)

In the above expressions, ĥ_(2e)(z) and ĥ_(2o)(z) represent thez-transforms of the even and odd components h_(2e) and h_(2o) of theinverse filter h₂, and ĝ_(2e)(z) and ĝ_(2o)(z) represent thez-transforms of the even and odd components g_(2e) and g_(2o) of theinverse filter g₂. Moreover, in the expressions of {circumflex over(x)}₁(z) and {circumflex over (x)}₂(z), ĥ′_(o)(z) represents thez-transform of the odd component h′_(o) of h′=h₂*h₁, and in theexpression of {circumflex over (x)}₃(z) and {circumflex over (x)}₄(z),ĝ′_(o)(z) represents the z-transform of the odd component g′_(o) ofg′=g₂*h₁:

${{\hat{h}}_{o}^{\prime}\left( z^{2} \right)} = {\frac{z}{2} \cdot \left( {{{{\hat{h}}_{2}(z)} \cdot {{\hat{h}}_{1}(z)}} - {{{\hat{h}}_{2}\left( {- z} \right)} \cdot {{\hat{h}}_{1}\left( {- z} \right)}}} \right)}$${{\hat{g}}_{o}^{\prime}\left( z^{2} \right)} = {\frac{z}{2} \cdot \left( {{{{\hat{g}}_{2}(z)} \cdot {{\hat{h}}_{1}(z)}} - {{{\hat{g}}_{2}\left( {- z} \right)} \cdot {{\hat{h}}_{1}\left( {- z} \right)}}} \right)}$

In the particular case of Haar filters, the resulting perfectreconstruction polyphase filters can be chosen as:

${{{\hat{x}}_{1}(z)} = {\frac{1}{2\sqrt{2}} \cdot \left( {1 + a + {\left( {1 - a} \right) \cdot z^{- 1}}} \right)}},{{{\hat{x}}_{2}(z)} = {\frac{1}{2\sqrt{2}} \cdot \left( {1 + a + {\left( {1 - a} \right) \cdot z}} \right)}},{{{\hat{x}}_{3}(z)} = {\frac{1}{2\sqrt{2}} \cdot \left( {1 + a - {\left( {1 - a} \right) \cdot z^{- 1}}} \right)}},{{{\hat{x}}_{4}(z)} = {\frac{1}{2\sqrt{2}} \cdot \left( {{- \left( {1 + a} \right)} + {\left( {1 - a} \right) \cdot z}} \right)}},{{{\hat{x}}_{5}(z)} = {{- \frac{1 - a}{\sqrt{2}}} \cdot z^{- 1}}},{{{\hat{x}}_{6}(z)} = {\frac{1 - a}{\sqrt{2}} \cdot z}}$

In a preferred embodiment, a mixing weight a=½ is chosen.

In the TOISF unit 700 of FIG. 7, the polyphase components of thehigh-pass input signal H are separated by the downsamplers 701, 703, thesamples submitted to downsampler 703 being previously shifted by −1 bythe shift module 702. In parallel, the low-pass signal L is input tofilters 704 and 705 which convolve it by x₁ and x₂, respectively. Theeven samples of H, output by downsampler 701, are input to filters 706and 707 which convolve them by x₃ and x₄, respectively. The odd samplesof H, output by downsampler 703, are input to filters 708 and 709 whichconvolve them by x₅ and x₆, respectively.

An adder 710 receives the respective outputs of filters 706 and 708, andits output is further added with the output of filter 704 (adder 711) toproduce the even components of the output signal Ā. Likewise, an adder712 receives the respective the respective outputs of filters 707 and709, and its output is further added with the output of filter 705(adder 713) to produce the odd components of the output signal Ā. Theeven and odd components of the output signal Ā are recombined byupsampling the even and odd components by a factor 2 (upsamplers 714,715), shifting by +1 the upsampled odd components (shift module 716) andadding the outputs of the upsampler 714 and of the shift module 716(adder 717), which yields the reconstructed signal Ā.

A TOSF filter bank 101, as used in the enhancement system of FIG. 1, canbe obtained from any critically-sampled filter bank by replacing thecritically-sampled subband filtering modules of FIG. 2 by TOSF units400, 600 as shown in either of FIG. 4 and FIG. 6. The TOISF filter bank103 is obtained from the corresponding critically-sampled inverse filterbank by replacing the critically-sampled inverse filtering modules ofFIG. 3 by TOISF units 500, 700 as shown in FIG. 5 or FIG. 7. A TOSFfilter bank may be regarded as a binary tree where each node is a TOSFunit that splits a signal into low- and high-frequency signals. Eachbinary tree corresponds to a particular wavelet packet transform.

FIG. 8 shows an exemplary embodiment of a TOSF filter bank for aone-dimensional input signal S[n] (d=1), in a case where the waveletpacket transform is a wavelet transform over three scales (J=3). Thefilter bank has J=3 modules 801, 802, 803 each consisting of 2^(d)−1=1TOSF filtering unit which may be designed according to either of FIG. 4and FIG. 6. The first TOSF unit 801 splits the input signal S into alow-frequency signal L having twice fewer samples than S and ahigh-frequency signal H=F₁ having the same number of samples as S. Thelow-frequency signal L is then decomposed by the next TOSF unit 802 thatis identical to 801. The TOSF unit 802 outputs a low-frequency signaland a high-frequency signal F₂. The low-frequency signal from the TOSFunit 802 is in turn decomposed by the last TOSF module 803, alsoidentical to 801, into a high-frequency signal F₃ and a low-frequencysignal F₄. The lowest-frequency output component F₄ has about 2^(d.J)=8times fewer samples than S; the highest-frequency output component F₁has the same number of samples as S; and the other output components F₂and F₃ respectively have oversampling ratios of 4 and 2 compared to F₄.

FIG. 9 shows an exemplary embodiment of a TOISF filter bank that invertsthe TOSF filter bank of FIG. 8. The TOISF filter bank receives theK=J+1=4 subband signals F ₁- F ₄ corresponding to the four componentsF₁-F₄ from the TOSF filter bank after processing in the enhancementmodule 102 of FIG. 1 (K=4 in this case). It has J=3 modules 901, 902,903 each consisting of 2^(d)−1=1 TOISF unit of the type shown in FIG. 5or 7. The TOISF units 901, 902 and 903 are the inverses of the TOSFunits 803, 802 and 801, respectively. The TOISF unit 901 takes in inputthe low-frequency signal F ₄ and the high-frequency signal F ₃ toreconstruct a signal that is the low-frequency signal L′ input to theTOISF unit 902, together with the high-frequency signal F ₂. The TOISFunit 902 outputs a reconstructed signal which, in turn, is thelow-frequency signal L input to the TOISF unit 903 together with thehigh-frequency signal F ₁. The TOISF unit 903 outputs the reconstructedsignal S.

FIG. 10 shows an exemplary embodiment of a TOSF filter bank for atwo-dimensional image signal S[n₁, n₂] (d=2), in a case where thewavelet packet transform is a wavelet transform over two scales (J=2).The integers n₁ and n₂ are respectively row and column indexes. Thefilter bank has J=2 modules 1000, 1010, hereafter referred to as “2DTOSF modules”, each consisting of 2^(d)−1=3 TOSF units 1001-1003,1004-1006 which may be designed according to either of FIG. 4 and FIG.6.

In the filter bank of FIG. 10, the input image S is supplied to the 2DTOSF module 1000 consisting of the three TOSF units 1001, 1002, 1003.The first TOSF unit 1001 applies the twice oversampled subband filteringto each row of the input image S. For each row, the TOSF unit 1001outputs a low-frequency signal L having twice fewer samples than S, anda high-frequency signal H having as many samples as S. The low-frequencysignal L for all rows defines a low-frequency image output that has asmany rows as S, but twice fewer columns. Each column of L is an inputsignal for the TOSF unit 1002. The high-frequency signal H for all rowsdefines a high-frequency image output that has as many rows and columnsas S. Each column of H is an input signal for the TOSF unit 1003.

For each column of H, the TOSF unit 1003 outputs a low-frequency signalhaving twice fewer samples than S and a high-frequency signal having asmany samples as S. The high-frequency signal for all columns defines asubband image output F₁ having as many rows and columns as S. Thelow-frequency signal from the TOSF unit 1003 for all columns defines asubband image output F₂ having as many columns as the input image S, buttwice fewer rows. The TOSF unit 1002 is identical to the TOSF unit 1003.Its high-frequency image output, having as many rows as S but twicefewer columns, is one of the subband components F₃ of the input image S.

The low-frequency image output L′ of the TOSF unit 1002 has twice fewerrows and columns than S. It is further decomposed by another 2D TOSFmodule 1010 having the same structure as the above-described 2D TOSFmodule 1000, with the three TOSF units 1004, 1005, 1006. The TOSF unit1004 again performs a twice oversampled subband filtering along rows toproduce a low-frequency image output L″ and high-frequency image outputH″. The two TOSF units 1005, 1006 are provided to further decompose therespective image outputs L″, H″ from unit 1004. The high- andlow-frequency image outputs from unit 1006 form two respective subbandcomponents F₄, F₅ of the input image S, with F₄ having twice fewer rowsand columns than S, and F₅ having twice fewer columns and four timesfewer rows than the input image S. The high- and low-frequency imageoutputs from unit 1006 also form two respective subband components F₆,F₇ of the input image S, with F₆ having twice fewer rows and four timesfewer columns than S, and F₇ having four times fewer rows and columnsthan the input image S.

The lowest-frequency output component F₇ in the example of FIG. 10 hasabout 2^(d.J)=16 times fewer samples than S; the highest-frequencyoutput component F₁ has the same number of samples as S; the outputcomponents F₂-F₃ have an oversampling ratio of 8 compared to F₇; theoutput component F₄ has an oversampling ratio of 4 compared to F₇; andthe output components F₅-F₆ have an oversampling ratio of 2 compared toF₇.

FIG. 11 shows an exemplary embodiment of a TOISF filter bank thatinverts the TOSF filter bank of FIG. 10. The TOISF filter bank receivesthe K=3J+1=7 subband signals F ₁- F ₇ corresponding to the sevencomponents F₁-F₇ from the TOSF filter bank after processing in theenhancement module 102 of FIG. 1 (K=7 in this case). It has J=2 modules1100, 1110, hereafter referred to as “2D TOISF modules”, each consistingof 2^(d)−1=3 TOISF units 1101-1103, 1104-1106 which may be of the typeshown in FIG. 5 or 7. The TOISF units 1101, 1102, 1103, 1104, 1105 and1106 are the inverses of the TOSF units 1006, 1005, 1004, 1003, 1002 and1001, respectively. In other words, the 2D TOISF module 1100 is theinverse of the 2D TOSF module 1010 of FIG. 10, while the 2D TOISF module1100 is the inverse of the 2D TOSF module 1000 of FIG. 10.

The TOISF unit 1102 takes in input the low-frequency signal F ₇ and thehigh-frequency signal F ₆ to reconstruct a signal that is thelow-frequency signal L″ input to the TOISF unit 1103. The TOISF unit1101 takes in input the high-frequency signal F ₄ and the low-frequencysignal F ₅ to reconstruct a signal that is the high-frequency signal H″input to the TOISF unit 1103, together with the low-frequency signal L″from the TOISF unit 1102. The TOISF unit 1103 outputs a low-frequencyreconstructed signal L′ which is the enhanced replica of thelow-frequency image L′ passed between the 2D TOSF units 1000, 1100 inFIG. 10. This low-frequency reconstructed signal L′ and the threeremaining components F ₁- F ₃ of the input of the TOISF filter bank aresupplied to the second 2D TOISF module 1100.

The TOISF unit 1105 takes in input the low-frequency reconstructedsignal L′ and the high-frequency signal F ₃ to reconstruct a signal thatis the low-frequency signal L input to the TOISF unit 1106. The TOISFunit 1104 takes in input the high-frequency signal F ₁ and thelow-frequency signal F ₂ to reconstruct a signal that is thehigh-frequency signal H input to the TOISF unit 1106, together with thelow-frequency signal L from the TOISF unit 1105. The TOISF unit 1106outputs the reconstructed signal S.

FIG. 12 shows an exemplary embodiment of a TOSF filter bank for athree-dimensional video signal S[n₁, n₂, n₃] (d=3), in a case where thewavelet packet transform is a wavelet transform over one scale (J=1). Tosimplify explanations, the integers n₁, n₂ and n₃ are respectivelyreferred to as row, column and time indexes (it may be noted that n₃ maynot correspond to a time parameter when the input signal is other than avideo). The filter bank has J=1 module 1200, hereafter referred to as“3D TOSF module”, consisting of 2^(d)−1=7 TOSF units 1201-1207 which maybe designed according to either of FIG. 4 and FIG. 6.

In the filter bank of FIG. 12, the input video signal S is supplied to a3D TOSF module 1200 consisting of three TOSF units 1201, 1202, 1203which may be designed according to either of FIG. 4 and FIG. 6. The TOSFunit 1201 applies the twice oversampled subband filtering to each row ofthe input signal S. For each row, the TOSF unit 1201 outputs alow-frequency signal L having twice fewer samples than S, and ahigh-frequency signal H having as many samples as S. The low-frequencysignal L for all rows defines a low-frequency signal output which issupplied to the TOSF unit 1202. The high-frequency signal H for all rowsdefines a high-frequency signal output which is supplied to the TOSFunit 1203. The TOSF units 1202, 1203 apply the twice oversampled subbandfiltering to each column of L and H, respectively. For each column ofH(L), the unit 1203 (1202) outputs a low-frequency signal L′₁ (L′₂)having twice fewer samples and a high-frequency signal H′₁ (H′₂) havingas many samples as H (L). The four signals L′₂, H′₂, L′₁ and H′₁ arefurther decomposed along the time dimension by the respective TOSF units1204, 1205, 1206 and 1207 to provide the subband components F₁-F₈.

The lowest-frequency output component F₈ in the example of FIG. 12 hasabout 2^(d.J)=8 times fewer samples than S; the highest-frequency outputcomponent F₁ has the same number of samples as S; the output componentsF₂-F₃ and F₅ have an oversampling ratio of 4 compared to F₈; and theoutput components F₄ and F₆-F₇ have an oversampling ratio of 2 comparedto F₈.

FIG. 13 shows an exemplary embodiment of a TOISF filter bank thatinverts the TOSF filter bank of FIG. 12. The TOISF filter bank receivesthe K=7J+1=8 subband signals F₁-F₈ corresponding to the eight componentsF₁-F₈ from the TOSF filter bank after processing in the enhancementmodule 102 of FIG. 1 (K=8 in this case). It has J=1 module 1300,hereafter referred to as “3D TOISF module”, consisting of 2^(d)−1=7TOISF units 1301-1307 which may be of the type shown in FIG. 5 or 7. TheTOISF units 1301, 1302, 1303, 1304, 1305, 1306 and 1307 are the inversesof the TOSF units 1207, 1206, 1205, 1204, 1203, 1202 and 1201,respectively. In other words, the 3D TOISF module 1300 is the inverse ofthe 3D TOSF module 1200 of FIG. 12.

The TOISF unit 1302 takes in input the low-frequency signal F ₄ and thehigh-frequency signal F ₃ to reconstruct a signal that is thelow-frequency signal L′₁ input to the TOISF unit 1305. The TOISF unit1301 takes in input the high-frequency signal F ₂ and the low-frequencysignal F ₁ to reconstruct a signal that is the high-frequency signal H′₁input to the TOISF unit 1305, together with the low-frequency signal L′₁from the TOISF unit 1302. The TOISF unit 1304 takes in input thelow-frequency signal F ₈ and the high-frequency signal F ₇ toreconstruct a signal that is the low-frequency signal L′₂ input to theTOISF unit 1306. The TOISF unit 1303 takes in input the high-frequencysignal F ₆ and the low-frequency signal F ₅ to reconstruct a signal thatis the high-frequency signal H′₂ input to the TOISF unit 1303, togetherwith the low-frequency signal L′₂ from the TOISF unit 1304. The TOISFunits 1305, 1306 output respective high- and low frequency reconstructedsignals H, L which are supplied to the final TOISF unit 1307 of the 3DTOISF module 1300. The TOISF unit 1307 outputs the reconstructed videosignal S.

The diagrams of FIGS. 8-13 are readily generalized to more than onescale and/or to more than three dimensions of the input signal S. For agiven signal dimension d, an additional scale J can be provided byfurther decomposing the lowest-frequency component of the decompositionat scale J−1 (F₄ in FIG. 8, F₇ in FIG. 10, F₈ in FIG. 12) by means of anadditional dD TOSF module (a 1D TOSF module consisting of just one TOSFunit as exemplified in FIG. 4 or 6). On the reconstruction side, afurther dD TOISF module is inserted at the input end of the filter bankto add one scale J (a 1D TOISF module consisting of just one TOISF unitas exemplified in FIG. 5 or 7). Each TOSF or TOISF module is organizedas a binary tree having d levels. One signal dimension is added bysimply adding one level to the binary tree.

For a d-dimensional signal decomposed with a wavelet transform up to ascale J:

-   -   a dD TOSF (TOISF) module has 2^(d) outputs (inputs);    -   the number of TOSF (TOISF) units in each dD TOSF (TOISF) module        is 2^(d)−1;    -   the number of dD TOSF modules for decomposition is J;    -   the number of 3D TOISF modules for reconstruction is also J; and    -   the number of decomposed signal components (outputs of the TOSF        filter bank 101/inputs of the TOISF filter bank 103) is        K=(2^(d)−1).J+1.

Various subband filtering modifications can be applied to theabove-described filter banks. For example, the TOSF units used in theTOSF filter bank 101 can be implemented with different filters h_(i),g_(i) at different levels of the decomposition tree. In such a case, thecorresponding TOISF filtering must use the corresponding pairs ofperfect reconstruction filters so that {h₁, g₁} and {h₂, g₂} have aperfect reconstruction property.

For videos (d=3), it may be worthwhile to use shorter filters along thetime dimension than along the spatial directions.

Other oversampled wavelet packet subband transforms that are not wavelettransforms can also be implemented in the TOSF filter bank 101 and inthe TOISF filter bank 103. At a scale J, the TOSF filter bank 101implementing an oversampled wavelet packet subband transform isorganized a tree of TOSF filtering modules of depth J. Each level j inthe tree has one module in the particular case of the wavelet transform(the J modules being cascaded as described above), or more than onemodule in the generalized wavelet packet case. The TOSF module at theroot of the tree decomposes the input signal S while each TOSF modulebeyond that sub-decomposes a component coming from a previous TOSFmodule that is not necessarily the low-frequency component.Symmetrically, the TOISF filter bank 103 reconstructs a signal byinverting the TOSF modules of the filter bank 101 with TOISF modulesorganized in an equivalent tree structure of the inverse filter bank ofdepth J.

FIG. 14 shows results of 2D image enhancement computed with theabove-described filter banks, compared with a conventional imageenhancement computed with a critically-sampled filter bank. FIG. 14( a)shows an original image. FIG. 14( b) shows the same image which has beenartificially corrupted by a Gaussian additive white noise with a PSNR28.46 dB. FIG. 14( d) shows an image obtained with an enhancement systemusing TOSF and TOISF filter banks implementing a twice oversampledtwo-dimensional wavelet transform over J=4 scales, using Haar filters,The subband enhancement is implemented with a thresholding operationsetting to zero all subband coefficients whose amplitude are larger than3 times the standard deviation of the noise. The image of FIG. 14( c) isobtained by replacing the twice oversampled Haar wavelet transform by anorthogonal Haar wavelet transform implemented with critically-sampledsubband filtering using the same Haar filters. It has a PSNR of 28.96 dBwith visible block or grid artifacts that have virtually disappearedwith the twice oversampled Haar wavelet transform for which the PSNR is31.16 dB.

1-34. (canceled)
 35. A filter bank for subband decomposition of an input signal into a number K of subband components for processing, comprising a filtering module for transforming the input signal into 2^(d) components including a low-frequency component and 2^(d)−1 higher-frequency components, d being an integer at least equal to 1 representing a dimension of the input signal, wherein said 2^(d)−1 higher-frequency components are oversampled compared to said low-frequency component.
 36. The filter bank as claimed in claim 35, wherein said low-frequency component is downsampled by a factor 2^(d) compared to the input signal, and wherein said 2^(d)−1 higher-frequency components include a highest-frequency component having as many samples as the input signal and, if d>1, further components oversampled by respective factors 2^(i) compared to said low-frequency component, each i being an integer greater than 0 and smaller than d.
 37. The filter bank as claimed in claim 35, wherein the K subband components include said 2^(d)−1 higher-frequency components.
 38. The filter bank as claimed in claim 37, wherein the subband decomposition is performed over one scale and the K subband components further include said low-frequency component.
 39. The filter bank as claimed in claim 35, wherein the subband decomposition is performed over multiple scales, the filter bank comprising at least J filtering modules organized in a tree structure having J levels, J being the number of scales of the subband decomposition, wherein each of the filtering modules is arranged to transform a respective input signal into 2^(d) respective components including a low-frequency component and 2^(d)−1 higher-frequency components oversampled compared to the respective low-frequency component, and wherein the input signal of the filter bank is the respective input signal of a filtering module at the root of the tree structure.
 40. The filter bank as claimed in claim 39, wherein the filter bank has one filtering module per level in the tree structure, and wherein, for j=1, . . . , J−1, the low-frequency component from the filtering module of the j-th level is the input signal to the filtering module of the (j+1)-th level.
 41. The filter bank as claimed in claim 40, wherein the K subband components include the low-frequency component from the filtering module of the J-th level and the 2^(d)−1 higher-frequency components from each one of the J filtering modules of the tree structure.
 42. The filter bank as claimed in claim 41, wherein the low-frequency component from the filtering module of the J-th level is downsampled by a factor 2^(d.J) compared to the input signal, and wherein the K subband components include: the low-frequency component from the filtering module of the J-th level; a highest-frequency component oversampled by a factor 2^(d) compared to said low-frequency component; and if at least one of d and J is greater than 1, further components oversampled by respective factors 2^(i) compared to said low-frequency component, each i being an integer greater than 0 and smaller than d.J.
 43. The filter bank as claimed in claim 35, wherein each filtering module is made of 2^(d)−1 filtering units arranged in a binary tree having d levels, wherein, for 1≦i≦d, the i-th level in the tree has 2^(i−1) filtering units each receiving a respective input signal and producing a respective output low-frequency signal and a respective output high-frequency signal oversampled compared to said respective output low-frequency signal, wherein the input signal of said filtering module is the input signal of the filtering unit of the first level in the tree, wherein, for any i>1, the output low-frequency and high-frequency signals from the 2^(i−2) filtering units of the (i−1)-th level in the tree are the respective input signals of the 2^(i−1) filtering units of the i-th level in the tree, and wherein the 2^(d) respective components from said filtering module are the output low-frequency and high-frequency signals from the 2^(d)−1 filtering units of the d-th level in the tree.
 44. The filter bank as claimed in claim 43, wherein each filtering unit has a first branch for transforming the respective input signal of the filtering unit into the respective output high-frequency signal and a second branch for transforming the respective input signal of the filtering unit into the respective output low-frequency signal downsampled by a factor of two compared to the respective input signal and the respective output high-frequency signal.
 45. The filter bank as claimed in claim 44, wherein the first branch comprises a high-pass filter and the second branch comprises a low-pass filter followed by a downsampler.
 46. The filter bank as claimed in claim 44, wherein the first branch comprises a high-pass filter and the second branch comprises a polyphase low-pass filtering arrangement.
 47. A filter bank for reconstruction of an output signal from a number K of processed subband components, comprising a filtering module for generating the output signal from 2^(d) components obtained from the K subband components, including a low-frequency component and 2^(d)−1 higher-frequency components, d being an integer at least equal to 1 representing a dimension of the output signal, wherein said 2^(d)−1 higher-frequency components are oversampled compared to said low-frequency component.
 48. The filter bank as claimed in claim 47, wherein the output signal is oversampled by a factor 2^(d) compared to said low-frequency component, and wherein said 2^(d)−1 higher-frequency components include a highest-frequency component having as many samples as the output signal and, if d>1, further components oversampled by respective factors 2^(i) compared to said low-frequency component, each i being an integer greater than 0 and smaller than d.
 49. The filter bank as claimed in claim 47, wherein said 2^(d)−1 higher-frequency components are components from said K subband components.
 50. The filter bank as claimed in claim 49, wherein the subband components result from a decomposition over one scale, and the K subband components further include said low-frequency component.
 51. The filter bank as claimed in claim 47, wherein the subband components result from a decomposition over multiple scales, the filter bank comprising J filtering modules organized in a tree structure having J levels, J being the number of scales of the decomposition, wherein each of the filtering modules is arranged to transform 2^(d) respective input components, including a low-frequency component and 2^(d)−1 higher-frequency components oversampled compared to the respective low-frequency component, into a respective output signal, wherein the output signal from a filtering module at the root of the tree structure is the output signal of the filter bank.
 52. The filter bank as claimed in claim 51, wherein the filter bank has one filtering module per level in the tree structure, and wherein, for j=1, . . . , J−1, the output signal from the filtering module of the j-th level is the respective low-frequency component supplied to the filtering module of the (j+1)-th level.
 53. The filter bank as claimed in claim 52, wherein the 2^(d)−1 higher-frequency components supplied to each one of the J filtering modules of the tree structure and the low-frequency component supplied to the filtering module of the first level are components from said K subband components.
 54. The filter bank as claimed in claim 53, wherein the output signal is oversampled by a factor 2^(d.J) compared to the low-frequency component supplied to the filtering module of the first level of the tree structure, and wherein the K subband components include: the low-frequency component supplied to the filtering module of the first level; a highest-frequency component oversampled by a factor 2^(d) compared to said low-frequency component; and if at least one of d and J is greater than 1, further components oversampled by respective factors 2^(i) compared to said low-frequency component, each i being an integer greater than 0 and smaller than d.J.
 55. The filter bank as claimed in claim 47, wherein each filtering module is made of 2^(d)−1 filtering units arranged in a binary tree having d levels, wherein, for 1≦i≦d, the i-th level in the tree has 2^(d−i) filtering units each receiving a respective input low-frequency signal and a respective input high-frequency signal oversampled compared to said respective input low-frequency signal and producing a respective output signal, wherein the 2^(d) respective input components supplied to said filtering module are distributed as respective input low-frequency and high-frequency signals to the 2^(d−1) filtering units of the first level in the tree, wherein, for any i>1, the respective input low-frequency and high-frequency signals of the 2^(d−i) filtering units of the i-th level in the tree are the respective output signals from the 2^(d−i+1) filtering units of the (i−1)-th level in the tree, and wherein the output signal of said filtering module is the output signal of the filtering unit of the d-th level in the tree.
 56. The filter bank as claimed in claim 55, wherein at least one of the filtering units has: a first branch for filtering an upsampled version of the respective input low-frequency signal of the filtering unit to form a first component signal; a second branch for filtering a first modified version of the respective input high-frequency signal of the filtering unit, in which the odd samples are replaced by zeroes, to form a second component signal; a first adder to form a first partially reconstructed signal as a sum of the first and second component signals; a third branch for filtering the partially reconstructed signal to form a third component signal; a fourth branch for filtering a second modified version of the respective input high-frequency signal of the filtering unit, in which the even samples are replaced by zeroes, to form a fourth component signal; a second adder to form a second partially reconstructed signal as a sum of the third and fourth component signals; and a combiner to produce the respective output signal of the filtering unit as a combination of the first and second partially reconstructed signals.
 57. The filter bank as claimed in claim 56, wherein the combination performed by the combiner is a weighted sum of the first and second partially reconstructed signals.
 58. The filter bank as claimed in claim 57, wherein the weighted sum of the first and second partially reconstructed signals is performed with equal weight coefficients.
 59. The filter bank as claimed in claim 56, wherein, the K processed subband components being obtained from a subband decomposition involving a low-pass filter h₁ and a high-pass filter g₁, the filtering performed in said first and second branches is based on respective filters h₂ and g₂ such that the filter pairs {h₁, g₁} and {h₂, g₂} verify a perfect reconstruction property, wherein the filtering performed in said third branch is based on a combination of filters h₁ and h₂, and wherein the filtering performed in said fourth branch is also based on filter g₂.
 60. The filter bank as claimed in claim 55, wherein at least one of the filtering units has: a first filter for generating a first even sub-component from the respective input low-frequency signal of the filtering unit; a second filter for generating a first odd sub-component from the respective input low-frequency signal of the filtering unit; a third filter for generating a second even sub-component from a first downsampled version of the respective input high-frequency signal of the filtering unit; a fourth filter for generating a second odd sub-component from said first downsampled version of the input high-frequency signal; a fifth filter for generating a third even sub-component from a second downsampled version of the respective input high-frequency signal of the filtering unit; a sixth filter for generating a third odd sub-component from said second downsampled version of the input high-frequency signal; a combiner to produce the respective output signal of the filtering unit having even components respectively obtained as a sum of the first, second and third even sub-components and as a sum of the first, second and third odd sub-components.
 61. The filter bank as claimed in claim 60, wherein, the K processed subband components being obtained from a subband decomposition involving a high-pass filter g₁ and a low-pass filter h₁ implemented in a polyphase filtering arrangement, said first, second, third, fourth, fifth and sixth filters are defined from said low-pass and high-pass filters h₁, g₁ and from respective inverse filters h₂ and g₂ such that the filter pairs {h₁, g₁ } and {h₂, g₂} verify a perfect reconstruction property, wherein: the first filter has a z-transform {circumflex over (x)} ₁(z)=a.ĥ _(2e)(z)+(1−a).z ⁻¹ .ĥ _(2o)(z).ĥ′ _(o)(z) the second filter has a z-transform {circumflex over (x)} ₂(z)=a.ĥ _(2o)(z)+(1−a).ĥ _(2e)(z).ĥ _(2e)(z).ĥ′ _(o)(z) the third filter has a z-transform {circumflex over (x)} ₃(z)=a.ĝ _(2e)(z)+(1−a)⁻¹ .ĥ ^(2o)(z).ĝ′ _(o)(z) the fourth filter has a z-transform {circumflex over (x)} ₄(z)=a.ĝ _(2o)(z)+(1−a).ĥ _(2e)(z).ĝ′ _(o)(z) the fifth filter has a z-transform {circumflex over (x)} ₅(z)=(1−a).z ⁻¹ .ĝ _(2o)(z); and the sixth filter has a z-transform {circumflex over (x)} ₆(z)=(1−a).ĝ _(2e)(z), where ĥ_(2e)(z) and ĥ_(2o)(z) represent the z-transforms of the even and odd components h_(2e) and h_(2o) of the inverse filter h₂, respectively, ĝ_(2e)(z) and ĝ_(2o)(z) represent the z-transforms of the even and odd components g_(2e) and g_(2o) of the inverse filter g₂, respectively, ĥ′_(o)(z) and ĝ′_(o)(z) represent the z-transforms of the odd components of h′=h₂*h₁ and of g′=g₂*h₁, respectively, where * designates convolution, and where a designates a weighting coefficient selected larger than 0 and strictly smaller than
 1. 62. The filter bank as claimed in claim 61, wherein a=½.
 63. The filter bank as claimed in claim 61, wherein the filters h₁, g₁, h₂ and g₂ are Haar wavelet filters having z-transforms ĥ₁(z)=(1+z)/√{square root over (2)}, ĝ₁(z)=(1−z)/√{square root over (2)}, ĥ₂(z)=(1+z⁻¹)/√{square root over (2)} and ĝ₂(z)=(1−z⁻¹)/√{square root over (2)}, and wherein the z-transforms of said first, second, third, fourth, fifth and sixth filters are defined as: ${{{\hat{x}}_{1}(z)} = {\frac{1}{2\sqrt{2}} \cdot \left( {1 + a + {\left( {1 - a} \right) \cdot z^{- 1}}} \right)}},{{{\hat{x}}_{2}(z)} = {\frac{1}{2\sqrt{2}} \cdot \left( {1 + a + {\left( {1 - a} \right) \cdot z}} \right)}},{{{\hat{x}}_{3}(z)} = {\frac{1}{2\sqrt{2}} \cdot \left( {1 + a - {\left( {1 - a} \right) \cdot z^{- 1}}} \right)}},{{{\hat{x}}_{4}(z)} = {\frac{1}{2\sqrt{2}} \cdot \left( {{- \left( {1 + a} \right)} + {\left( {1 - a} \right) \cdot z}} \right)}},{{{\hat{x}}_{5}(z)} = {{- \frac{1 - a}{\sqrt{2}}} \cdot z^{- 1}}},{{{and}\mspace{14mu} {{\hat{x}}_{6}(z)}} = {\frac{1 - a}{\sqrt{2}} \cdot {z.}}}$
 64. An enhancement system for d-dimensional signals, d being an integer at least equal to 1, comprising: a first filter bank for subband decomposition of an input signal into a number K of subband components; a subband enhancement module for processing the K subband components from the first filter bank and forming K enhanced subband components; and a second filter bank for reconstruction of an output signal from the K enhanced subband components, wherein the first filter bank comprises a first filtering module for transforming the input signal into 2^(d) components including a first low-frequency component and 2^(d)−1 first higher-frequency components oversampled compared to said first low-frequency component, and wherein the second filter bank comprises a second filtering module for generating the output signal from 2^(d) components obtained from the K enhanced subband components, including a second low-frequency component and 2^(d)−1 second higher-frequency components oversampled compared to said second low-frequency component.
 65. The enhancement system as claimed in claim 64, wherein the input and output signals are two-dimensional image signals.
 66. The enhancement system as claimed in claim 64, wherein the input and output signals are three-dimensional video signals.
 67. The enhancement system as claimed in claim 64, wherein the subband decomposition is based on wavelet transforms.
 68. The enhancement system as claimed in claim 64, wherein the subband enhancement module is arranged to perform a processing selected from thresholding, reduction of compression distortion, reduction of measurement noise and sharpness enhancement. 